In the problem of online load balancing on uniformly related machines with bounded migration, jobs arrive online one after another and have to be immediately placed on one of a given set of machines without knowledge about jobs that may arrive later on. Each job has a size and each machine has a speed, and the load due to a job assigned to a machine is obtained by dividing the first value by the second. The goal is to minimize the maximum overall load any machine receives. However, unlike in the pure online case, each time a new job arrives it contributes a migration potential equal to the product of its size and a certain migration factor. This potential can be spend to reassign jobs either right away (non-amortized case) or at any later time (amortized case). Semi-online models of this flavor have been studied intensively for several fundamental problems, e.g., load balancing on identical machines and bin packing, but uniformly related machines have not been considered up to now. In the present paper, the classical doubling strategy on uniformly related machines is combined with migration to achieve an $(8/3+\varepsilon)$-competitive algorithm and a $(4+\varepsilon)$-competitive algorithm with $O(1/\varepsilon)$ amortized and non-amortized migration, respectively, while the best known competitive ratio in the pure online setting is roughly $5.828$.